where \(\hat{\mathbf {S}}(k,:)\)  is the \(k\) th row of the estimation of device state matrix, i.e., \(\hat{\mathbf {S}}\) , \(v^2r^e\)  is the threshold, and \(v=v_1\max (|\hat{s}_{nr}|), \forall n\in \lbrace 1,\cdots ,N\rbrace ,\forall r \in \lbrace 1,\cdots ,r^e\rbrace \)  with \(\max (|\hat{s}_{nr}|)\)  being the maximum absolute value of element of \(\hat{\mathbf {S}}\) . Herein, \(v_1\)  is set according to the considered channel model. In particular, \(v_1\)  is set as the ratio of the minimum and the maximum amplitudes of the generated channel coefficients, which has been widely adopted in [1]} and [2]}. The threshold in (REF ) implies that if the energy of \(\hat{\mathbf {S}}(k,:)\)  is not less than the \(r^e\)  fold of the minimum element energy in statistical sense, the \(k\) th device is declared active.
